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Graphical Examination of Data. 1.12.1999 Jaakko Leppänen [email protected] Sources. H. Anderson, T. Black: Multivariate Data Analysis, (5th ed., p.40-46) . Yi-tzuu Chien: Interactive Pattern Recognition, (Chapter 3.4) .

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Graphical examination of data

Graphical Examination of Data

1.12.1999

Jaakko Leppänen

[email protected]


Sources
Sources

  • H. Anderson, T. Black: Multivariate Data Analysis,(5th ed., p.40-46).

  • Yi-tzuu Chien: Interactive Pattern Recognition,(Chapter 3.4).

  • S. Mustonen: Tilastolliset monimuuttujamenetelmät,(Chapter 1, Helsinki 1995).


Agenda
Agenda

  • Examining one variable

  • Examining the relationship between two variables

  • 3D visualization

  • Visualizing multidimensional data


Examining one variable
Examining one variable

  • Histogram

    • Represents the frequency of occurences within data categories

      • one value (for discrete variable)

      • an interval (for continuous variable)


Examining one variable1
Examining one variable

  • Stem and leaf diagram (A&B)

    • Presents the same graphical information as histogram

    • provides also an enumeration of the actual data values


Examining the relationship between two variables
Examining the relationship between two variables

  • Scatterplot

    • Relationship of two variables

Linear

Non-linear

No correlation


Examining the relationship between two variables1
Examining the relationship between two variables

  • Boxplot (according A&B)

    • Representation of data distribution

    • Shows:

      • Middle 50% distribution

      • Median (skewness)

      • Whiskers

      • Outliers

      • Extreme values


3d visualization
3D visualization

  • Good if there are just 3 variables

  • Mustonen: “Problems will arise when we should show lots of dimensions at the same time. Spinning 3D-images or stereo image pairs give us no help with them.”


Visualizing multidimensional data
Visualizing multidimensional data

  • Scatterplot with varying dots

  • Scatterplot matrix

  • Multivariate profiles

  • Star picture

  • Andrews’ Fourier transformations

  • Metroglyphs (Anderson)

  • Chernoff’s faces


Scatterplot
Scatterplot

  • Two variables for x- and y-axis

  • Other variables can be represented by

    • dot size, square size

    • height of rectangle

    • width of rectangle

    • color


Scatterplot matrix
Scatterplot matrix

  • Also named as Draftsman’s display

  • Histograms on diagonal

  • Scatterplot on lower portion

  • Correlations on upper portion


Scatterplot matrix cont
Scatterplot matrix (cont…)

correlations

histograms

scatterplots


Scatterplot matrix cont1
Scatterplot matrix (cont…)

  • Shows relations between each variable pair

  • Does not determine common distribution exactly

  • A good mean to learn new material

  • Helps when finding variable transformations


Scatterplot matrix as rasterplot
Scatterplot matrix as rasterplot

  • Color level represents the value

    • e.g. values are mapped to gray levels 0-255


Multivariate profiles
Multivariate profiles

  • A&B: ”The objective of the multivariate profiles is to portray the data in a manner that enables each identification of differences and similarities.”

  • Line diagram

    • Variables on x-axis

    • Scaled (or mapped) values on y-axis


Multivariate profiles cont
Multivariate profiles (cont…)

  • An own diagram for each measurement (or measurement group)


Star picture
Star picture

  • Like multivariate profile, but drawn from a point instead of x-axis

  • Vectors have constant angle


Andrews fourier transformations
Andrews’ Fourier transformations

  • D.F. Andrews, 1972.

  • Each measurement X = (X1, X2,..., Xp) is represented by the function below, where - < t < .


Andrews fourier transformations cont
Andrews’ Fourier transformations (cont…)

  • If severeal measurements are put into the same diagram similar measurements are close to each other.

  • The distance of curves is the Euklidean distance in p-dim space

  • Variables should be ordered by importance



Andrews fourier transformations cont2
Andrews’ Fourier transformations (cont…)

  • Can be drawn also using polar coordinates


Metroglyphs andersson
Metroglyphs (Andersson)

  • Each data vector (X) is symbolically represented by a metroglyph

  • Consists of a circle and set of h rays to the h variables of X.

  • The lenght of the ray represents the value of variable


Metroglyphs cont
Metroglyphs (cont...)

  • Normally rays should be placed at easily visualized and remembered positions

  • Can be slant in the same direction

    • the better way if there is a large number of metrogyphs


Metroglyphs cont1
Metroglyphs (cont...)

  • Theoretically no limit to the number of vectors

  • In practice, human eye works most efficiently with no more than 3-7 rays

  • Metroglyphs can be put into scatter diagram => removes 2 vectors


Chernoff s faces
Chernoff’s faces

  • H. Chernoff, 1973

  • Based on the idea that people can detect and remember faces very well

  • Variables determine the face features with linear transformation

  • Mustonen: "Funny idea, but not used in practice."


Chernoff s faces cont
Chernoff’s faces (cont…)

  • Originally 18 features

    • Radius to corner of face OP

    • Angle of OP to horizontal

    • Vertical size of face OU

    • Eccentricity of upper face

    • Eccentricity of lower face

    • Length of nose

    • Vertical position of mouth

    • Curvature of mouth 1/R

    • Width of mouth


Chernoff s faces cont1
Chernoff’s faces (cont…)

  • Face features (cont…)

    • Vertical position of eyes

    • Separation of eyes

    • Slant of eyes

    • Eccentricity of eyes

    • Size of eyes

    • Position of pupils

    • Vertical position of eyebrows

    • Slant of eyebrows

    • Size of eyebrows



Conclusion
Conclusion

  • Graphical Examination eases the understanding of variable relationships

  • Mustonen: "Even badly designed image is easier to understand than data matrix.”

  • "A picture is worth of a thousand words”


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